It is known for a finitely connected domain with \(C^{\infty }\)  boundary that the analytic capacity and (on-diagonal) Szegö kernel satisfy the relation \(c_B(z) = 2\pi S(z)\) . For \(\lbrace \Omega _j\rbrace _{j=1}^{\infty }\)  given above, as shown by Ahlfors and Beurling [1]}, the analytic capacities \(c_{j; B}\)  and \(c_B\)  of these domains, respectively, similarly satisfy
\(\lim _{j \rightarrow \infty } c_{j; B}(z) = c_B(z), \quad z \in \Omega .\) 
